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Quantum field theory and composite fermions in the fractional quantum Hall effect
Article first published online: 5 MAY 2009
Copyright © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Annalen der Physik
Volume 18, Issue 5, pages 285–377, May 2009
How to Cite
Kossow, M. (2009), Quantum field theory and composite fermions in the fractional quantum Hall effect. Ann. Phys., 18: 285–377. doi: 10.1002/andp.200910350
- Issue published online: 5 MAY 2009
- Article first published online: 5 MAY 2009
- Manuscript Accepted: 17 MAR 2009
- Manuscript Received: 4 MAR 2009
- Composite fermions;
- fractional quantum Hall effect;
- Chern Simons theory.
We derive a composite fermion model for the fractional quantum Hall effect from relativistic quantum electrodynamics. With simple arguments it is shown that a special Chern Simons transformation of the Dirac electrons in four spacetime dimensions leads in the low energy limit to a single particle Hamiltonian for composite fermions in three dimensions with correction terms such as Rashba- or Dresselhaus-spin-orbit coupling and zitterbewegung. Furthermore we provide a mechanism to quantum-mechanically project the quantum fields defined in the four dimensional Minkowski space to three dimensions. This leads to a relativistic field theory and especially a composite fermion field theory in three dimension. This projection map can be combined with the projection onto a Landau level or composite fermion Landau level respectively. This results in a quasi relativistic quantum field theory on a noncommutative plane. The phenomenological models resulting from this approach are discussed and allow a systematical exploration of the effects of the spin and the condensation in a Landau level. We expect from the relativistic approach corrections in terms of spin-orbit coupling effects. From the projection onto Landau levels we expect a modification of the dispersion relation and a modified composite fermion mass. Furthermore, the BRST quantization for Chern Simons theories with compact gauge group is reviewed and the phenomenological consequences within a composite fermion model with spin are discussed. The connection to Wess Zumino Witten theories is recalled and a possible link between the corresponding central charge of the related affine Lie algebra and the composite fermion filling factor is pointed out.